Something I've always liked doing is collecting cards such as in the pokemon TCG on the gameboy. These could be dropped from wild pokemon, obtained from booster packs with random ones or prizes. Gives you something extra to aim for in the form of rare cards, card battles, card tournaments and its like a game within a game.
The cards would just have a stat on each side and be used in a simple triple triad game made popular by the game in final fantasy 8. This is played out on a 3 by 3 grid. Higher numbers wins the sides it faces with lower numbers as each player plays 1 card at a time.
https://pokemonessentials.wikia.com/wiki/Triple_Triad
Although this game seems to have plenty of features already I think this could be a cool addition